Rectangles & Area Calculations using Algebra

Math Problem Solving with Algebra: Find the area of the rectangle Q when the areas of rectangles P, R & S are 56 sq.cm, 21 sq.cm & 48 sq.cm respectively.

Solving: Lets create a diagram (A) like below &

• Let us mark all the lines & the intersections for these 4 rectangles with alphabets E, F, G, H, I, J, K, L & M
• Let us mark the unique lengths with variables a, b, c & d

To solve this math puzzle, you would need to know:

1. How to calculate the area of a rectangle & the formula
2. Multiplication of fractions

From the diagram (A), we know:

For rectangle P:

1. Area of rectangle P = 56 sq.cm
2. P has sides HE, EF, FI & IH
3. Lets assume: HE = a cm, EF = c cm, FI = a cm (since it must be equal to the other parallel side HW) & IH = c cm (since it must be equal to the other parallel side EF)
4. So HE = FI = a & EF = IH = c
5. Since Area = HE * EF,  a*c = 56

For rectangle R:

6. Area of rectangle R = 21 sq.cm
7. R has sides KH, HI, IL & LK
8. Lets assume: KH = b cm, HI = c cm, IL = b cm (since it must be equal to the other parallel side KH) & LK = c cm (since it must be equal to the other parallel side HI)
9. So KH = IL = b & HI = LK = c
10. Since Area = KH * HI,  b*c = 21

For rectangle S:

11. Area of rectangle S = 48 sq.cm
12. S has sides LI, IJ, JM & ML
13. Lets assume: LI = b cm, IJ = d cm, JM = b cm (since it must be equal to the other parallel side LI) & ML = d cm (since it must be equal to the other parallel side IJ)
14. So LI = JM = b & IJ = ML = d 
15. Since Area = JM * ML,  b*d = 48

Area of rectangle Q IS UNKNOWN & let us say it is X sq.cm 

Area of Q = Length of Q * Width of Q

=> X = IF * FG 

=> X = a * d

Multiplying both numerator & denominator by ( b * c ) & rearranging the numbers:

⇒ X = (a * d * b * c) / (b * c)

=> X = (a * c) * (b * d) / (b * c) —————> (16)

From (5): a * c = 56

From (10): b * c = 21

From (15): b * d = 48

Putting these values in (16) & simplifying:

X = 56 * 48 / 21 = 128 sq.cm

Answer: So the Area of rectangle Q = 128 sq.cm